Is it possible that every class has a higher recall than precision for multi-class classification?
In the case of binary classification, i.e. when you have two classes $A$ and $B$, the recall is defined as
$${\displaystyle {\text{Recall}}={\frac {tp}{tp+fn}}}$$
Similarly, the precision as
$${\displaystyle {\text{Precision}}={\frac {tp}{tp+fp}}}$$
where, if you choose and fix a class $A$ as your main class (or class of interest),

$tp$ is the number of true positives (i.e. the number of observations that have been classified as $A$ and actually belong to class $A$, i.e. they have been correctly classified)

$fn$ is the number of false negatives (i.e. the number of observations that have been incorrectly classified as $B$, so, in this case of binary classification, they actually belong to class $A$)

$fp$ is the number of false positives (i.e. the number of observations that have been incorrectly classified as $A$)


In this context, the confusion matrix will be a $2 \times 2$ matrix because you have two predictions (one for each of the two classes) compared against the two ground-truth labels (or classes).
In the context of multi-class classification, where you have $N &gt; 2$ classes, you will have a $N \times N$ confusion matrix, for the same reason. In this case, the recall and precision are just an extension of the binary classification problem. More precisely, let $M$ represent your confusion matrix, so $M_{ij}$ is the entry of this confusion matrix at row $i$ and column $j$. Then the recall, for class $i$ (i.e. your main class, as $A$ above), is defined as
$$\text{Recall}_{~i} = \cfrac{M_{ii}}{\sum_j M_{ij}}$$
$$\text{Precision}_{~i} = \cfrac{M_{ii}}{\sum_j M_{ji}}$$
In other words, the recall for a certain class $i$ is computed as the fraction of the diagonal element $M_{ii}$ and the sum of all elements in that row $i$. Similarly, the precision is computed as the fraction of $M_{ii}$ and the sum of all elements in column $i$.
If the true positives (i.e. the diagonal elements $M_{ii}$) of all classes are indeed bigger compared to the denominator, the recall can indeed be bigger than the precision for all classes.